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Hurwitz quaternions are defined as:

$$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$ (that is, all integer or half integer quaternions).

I'm trying to show that if $a$ is a unit, then $||a||=1$ (the usual quaternion norm), but I can only sort of show that using very tedious calculations. Is there a short(er) and more elegant way to see this, preferably purely algebraic?

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up vote 7 down vote accepted

Suppose $x$ is a unit. Then $x^{-1}$ satisfies $\lVert x \rVert \lVert x^{-1} \rVert = 1$. So $\lVert x^{-1} \rVert = \lVert x \rVert ^{-1}$, and in particular one of $\lVert x \rVert$, $\lVert x^{-1} \rVert$ must be less than or equal to 1. But it's not hard to see that in fact the norm is an integer and not zero, so it must be 1.

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